A Bayesian formulation of hybrid quantum-classical dynamics

Abstract

We develop a Bayesian formulation of diffusive quantum-classical dynamics by treating the wave function and classical variables as components of an ordinary stochastic process. The joint probability density P(ψ,x,t) obeys a classical Fokker-Planck equation, while the quantum state appears as its second moment. Requiring this second moment to evolve linearly and autonomously yields the hybrid Lindblad equation and its stochastic unravelings. This construction makes positivity and unraveling freedom immediate and gives a unified description of quantum noise, classical noise, and their correlations through the covariance matrices (C,Γ,Q). The same stochastic representation turns quantum-classical state estimation into a classical hidden-state inference problem. Filtering and smoothing are Bayesian conditioning on the observed classical trajectory. We recover the stochastic master equation from the Kushner-Stratonovich equation with correlated noise and show how the quantum effect operator is related to the Bayesian backward message through the adjoint dynamics of the linear unraveling. The Bayesian posterior also defines a smoothed density matrix and, more generally, a posterior distribution over latent quantum-classical trajectories. These quantities can be approximated with standard particle filtering and smoothing methods. Numerical examples show that smoothing improves reconstruction of a hidden quantum-classical trajectory and that the full trajectory posterior can retain structure, such as multimodality, that is absent from its density-matrix second moment. The resulting framework connects quantum filtering, retrodiction, and smoothing to the standard forward-backward machinery of Bayesian time-series inference.